Why Does the World Exist?: An Existential Detective Story (22 page)

So he was more than a Platonist—he was a Pythagorean! Or he was at least flirting with Pythagoras’s mystical doctrine that the world was constituted by mathematics:
all is number
. Yet I noticed that there was one link among his three worlds that Penrose hadn’t yet addressed. He had mentioned how the mental world might be linked to the physical world, and how the physical world might be linked to the Platonic world of abstract mathematical ideas. But what about the supposed link between that Platonic world and the mental world? How are our minds supposed to get in touch with these incorporeal Platonic Forms? If we are to have knowledge of mathematical entities, we have to “perceive” them in some way, as Gödel put it. And perceiving an object usually means having causal commerce with it. I perceive the cat on the mat, for example, because photons emitted from the cat are impinging on the retinas of my eyes. But Platonic Forms aren’t like the cat on the mat. They don’t live in the world of space and time. There are no photons traveling back and forth between them and us. So we can’t perceive them. And if we can’t perceive mathematical objects, how can we possibly come to have knowledge of them?

Plato believed that such knowledge was derived from a previous existence, before we were born, during which our souls communed directly with the Forms; what we know of mathematics—and of Beauty and Goodness, for that matter—thus consists of “reminiscences” from this disembodied existence that preceded our earthly lives. Nobody takes that idea seriously any more. Yet what’s the alternative? Penrose himself had written that human consciousness somehow “breaks through” to the Platonic world when we contemplate mathematical objects. But consciousness depends on physical processes in the brain, and it’s hard to see how such processes could be affected by a nonphysical reality.

When I put this objection to Penrose, he furrowed his brow and was silent for a moment. “I know that’s something philosophers worry about,” he said at length. “But I’m not sure I’ve ever really understood the argument. It’s
out there
, the Platonic world, and we
can
have access to it. Ultimately, our physical brains are constructed out of material that is itself intimately related to the Platonic world of mathematics.”

So was he saying that we can perceive mathematical reality because our brains are somehow a
part
of that reality?

“It’s a little more complicated than that,” Sir Roger corrected me. “Each of the three worlds—the physical world, the world of consciousness, and the Platonic world—emerges out of a tiny bit of one of the others. And it’s always the most
perfect
bit. Consider the human brain. If you look at the entire physical cosmos, our brains are a tiny, tiny part of it. But they’re the most perfectly organized part. Compared to the complexity of a brain, a galaxy is just an inert lump. The brain is the most exquisite bit of physical reality, and it’s just this bit that gives rise to the mental world, the world of conscious thought. Likewise, it’s only a small part of our conscious thought that connects us to the Platonic world, but it’s the purest part—the part that consists in our contemplation of mathematical truth. Finally, just a few bits of the mathematics in the Platonic world are needed to describe the entire physical world—but they’re quite the most powerful and extraordinary bits!”

Spoken like a true mathematical physicist, I thought to myself. But could it be that these “powerful and extraordinary” bits of mathematics—the bits that preoccupy Penrose—are so powerful that they can generate a physical world all by themselves? Does mathematics carry its own ontological clout?

“Something like that, yes,” Sir Roger said. “Maybe philosophers worry too much about lesser issues without realizing that this is perhaps the greatest mystery of all: how the Platonic world ‘controls’ the physical world.”

He paused for a moment to reflect, and then added, “I’m not saying I can
resolve
this mystery.”

After a bit of small talk about Gödel’s incompleteness theorems, quantum computation, artificial intelligence, and animal consciousness (“I have no idea whether a starfish is conscious,” Penrose said, “but there
should
be some observable signs”), my visit with Sir Roger came to an end. I left his penthouse world of Platonic ideas and, after a quick elevator descent, reentered the ephemeral world of sensuous appearances below. Retracing my steps through Washington Square, I walked under the “hanging tree,” past the chess hustlers, and into the crowded plaza around the central fountain, encountering the same chaos of exuberant motion, garish colors, pungent smells, and exotic noises. These people! I thought, What do they know of the serene and timeless Platonic realm? Whether they be tourists or buskers, panhandlers or adolescent anarchists, or even NYU professors of cultural studies taking a shortcut through the square on their way to a lecture, their consciousness never touches the ethereal realm of mathematical abstraction that is the true source of reality. Little did they realize that, despite the abundant sunshine, they were chained in the allegorical darkness of Plato’s cave, condemned to live in a world of shadows. They could have no genuine knowledge of reality. That was open only to those who apprehended the eternal Forms, the true philosophers—like Penrose.

But gradually the spell that Sir Roger had cast on me began to wear off. How could the solemn mathematical abstractions in Plato’s heaven have given rise to the gaiety of life in Washington Square? Do such abstractions really hold the answer to the mystery of why there is Something rather than Nothing?

The scheme of being that Penrose had conjured up for me seemed almost miraculously self-creating and self-sustaining. There are three worlds: the Platonic world, the physical world, and the mental world. And each of the worlds somehow engenders one of the others. The Platonic world, through the magic of mathematics, engenders the physical world. The physical world, through the magic of brain chemistry, engenders the mental world. And the mental world, through the magic of conscious intuition, engenders the Platonic world—which, in turn, engenders the physical world, which engenders the mental world, and so on, around and around. Through this self-contained causal loop—Math creates Matter, Matter creates Mind, and Mind creates Math—the three worlds mutually support one another, hovering in midair over the abyss of Nothingness, like one of Penrose’s impossible objects.

Yet, despite what this picture might suggest, the three worlds are not ontologically coequal. It is the Platonic world, in Penrose’s vision, that is the
fons et origo
of reality. “
To me the world
of perfect forms is primary, its existence being almost a logical necessity—and
both
the other worlds are its shadows,” he wrote in
Shadows of the Mind
. The Platonic world, in other words, is compelled to exist by logic alone, and the contingent world—the world of matter and mind—follows as a shadowy by-product. That’s Penrose’s solution to the puzzle of existence.

And it left me with two misgivings. Is the existence of the Platonic world really assured by logic itself? And even if it is, what then makes it cast shadows?

As to the first, I couldn’t help noticing what looked like a failure of nerve on Penrose’s part. The existence of the Platonic world, he said, is “almost a logical necessity.” Why this “almost”? Logical necessity is not a thing that admits of degree. It is all or nothing. Penrose makes much of the alleged fact that the Platonic world of mathematics is “
eternally existing
,” that its reality is “
profound and timeless
.” But the same, one might note, would be true of God—
if
God existed. Yet God is not a logically necessary being; his existence can be denied without contradiction. Why should mathematical objects be superior to God in this respect?

The belief that the objects of pure mathematics exist necessarily has been called an “
ancient and honorable
” one, but it doesn’t hold up terribly well under scrutiny. It seems to be based on two premises: (1) mathematical truths are logically necessary; and (2) some of those truths assert the existence of abstract objects. As an example, consider proposition twenty in Euclid’s
Elements
, which says that there are infinitely many prime numbers. This certainly looks like an existence claim. Moreover, it appears to be true as a matter of logic. Indeed, Euclid proved that denying the existence of an infinity of primes led straight to absurdity. Suppose there were only finitely many prime numbers. Then, by multiplying them all together and adding 1, you would get a new number that was bigger than all the primes and yet divisible by none of them—contradiction!

Euclid’s
reductio ad absurdum
proof of the infinity of prime numbers has been called the first truly elegant bit of reasoning in the history of mathematics. But does it give any grounds for believing in the existence of numbers as eternal Platonic entities? Not really. In fact, the existence of numbers is
presupposed
by the proof. What Euclid really showed was that
if
there are infinitely many things that behave like the numbers 1, 2, 3, … ,
then
there must be infinitely many things among them that behave like
prime
numbers. All of mathematics can be seen to consist of such
if-then
propositions:
if
such-and-such a structure satisfies certain conditions,
then
that structure must satisfy certain further conditions. These
if-then
truths are indeed logically necessary. But they do not entail the existence of any objects, whether abstract or material. The proposition “2 + 2 = 4,” for example, tells you that
if
you had two unicorns and you added two more unicorns,
then
you would end up with four unicorns. But this
if-then
proposition is true even in a world that is devoid of unicorns—or, indeed, in a world that contains nothing at all.

Mathematicians essentially make up complex fictions. Some of these fictions have analogues in the physical world; they compose what we call “applied mathematics.” Others, like those positing higher infinities, are purely hypothetical. Mathematicians, in creating their imaginary universes, are constrained only by the need to be logically consistent—and to create something of beauty. (“
‘Imaginary universes’ are
so much more beautiful than the stupidly constructed ‘real’ one,” declared the great English number theorist G. H. Hardy.) As long as a collection of axioms does not lead to a contradiction, then it is at least
possible
that it describes something. That is why, in the words of Georg Cantor, who pioneered the theory of infinity, “
the essence of mathematics
is freedom.”

So the existence of mathematical objects is not mandated by logic, as Penrose seemed to believe. It is merely
permitted
by logic—a much weaker conclusion. Practically anything, after all, is permitted by logic. But for some modern-day Platonists of an even more radical stripe, that seems to be permission enough. As far as they are concerned, self-consistency alone guarantees mathematical existence. That is, as long as a set of axioms does not lead to a contradiction, then the world it describes is not only possible—it is actual.

One such radical Platonist is Max Tegmark, a young Swedish-American cosmologist who teaches at MIT. Tegmark believes, like Penrose, that the universe is inherently mathematical. Also like Penrose, he believes that mathematical entities are abstract and immutable. Where he goes beyond Sir Roger is in holding that
every
consistently describable mathematical structure exists in a genuine physical sense. Each of these abstract structures constitutes a parallel world, and together these parallel worlds make up a mathematical multiverse. “
The elements of this multiverse
do not reside in the same space but exist outside of space and time,” Tegmark has written. They can be thought of as “static sculptures that represent the mathematical structure of the physical laws that govern them.”

Tegmark’s extreme Platonism furnishes a very cheap resolution to the mystery of existence. It is basically, as he concedes, a mathematical version of Robert Nozick’s principle of fecundity, which says that reality encompasses all logical possibilities, that it is as rich and variegated as it can be. Anything that is possible must actually exist—hence the triumph of Something over Nothing. What makes such a principle compelling for Tegmark is the peculiar ontological muscle that mathematics seems to possess. Mathematical structures, he says, “
have an eerily real feel
to them.” They are fruitful in uncovenanted ways; they surprise us; they “bite back.” We get more out of them than we seem to have put into them. And if something
feels
so real, it must
be
real.

But why should we be swayed by this “real feel,” no matter how eerie? Tegmark and Penrose may be swayed, but another great physicist, Richard Feynman, was decidedly not. “
It’s just a feeling
,” Feynman once said dismissively, when asked whether the objects of mathematics had an independent existence.

Bertrand Russell came to take an even sterner view of such mathematical romanticism. In 1907, when he was in his relatively youthful thirties, Russell penned a gushing tribute to the transcendent glories of mathematics. “
Rightly viewed
,” he wrote, mathematics “possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture.” Yet by his late eighties he had come to view his callow rhapsodizing as “
largely nonsense
.” Mathematics, the aged Russell wrote, “has ceased to seem to me non-human in its subject-matter. I have come to believe, though very reluctantly, that it consists of tautologies. I fear that, to a mind of sufficient intellectual power, the whole of mathematics would appear trivial, as trivial as the statement that a four-footed animal is an animal.”